Statement: \(2m = m\): For all infinite cardinals \(m\), \(2m = m\).
Howard_Rubin_Number: 3
Parameter(s): This form does not depend on parameters
This form's transferability is: Unknown
This form's negation transferability is: Negation Transferable
Article Citations:
Tarski-1954a: Theorems on the existence of successors of cardinals and the axiom of choice
Book references
Note connections:
| Howard-Rubin Number | Statement | References |
|---|---|---|
| 3 A | For all cardinals \(m\) and \(n\), \(n\le m\) and \(m\)infinite \(\rightarrow m + n = m\). Halpern/Howard [1970]. |
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| 3 B | \(\aleph_{0}\cdot m = m\): For all infinite cardinals\(m\), \(\aleph_{0}\cdot m = m\). Halpern/Howard [1970]. |
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| 3 C | Definability of cardinal addition as the least upperbound: (For all cardinals \(x, y\) and \(z)( x + y = z\) iff \(( x \le z\)and \(y \le z\) and \((\forall u)( x,y \le u \rightarrow z \le u))\).H\"aussler [1983] and Tarski [1949a]. |
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| 3 D | Every pair of cardinals has a least upper bound inthe usual cardinal ordering and for all cardinals \(\kappa\) and \(\gamma\),if \(n\kappa \le \gamma \) for all \(n \in \omega \), then\(\aleph _{0}\kappa \le \gamma \). Hickman [1979b]. |
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| 3 E | \(E(IV,V)\): For all infinite cardinals \(m\), \(m + 1 = m\)implies \(2m= m\). Howard/Yorke [1989] and Note 94. |
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| 3 F | \(E(III,V)\): For every set \(X\), if \(\Cal P(X)\) isDedekind infinite then \(2|X| = |X|\). Howard/Yorke [1989] andNote 94. |
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| 3 G | \(E(II,V)\): For every set \(X\), if there is a non-emptyfamily of subsets of \(X\) which is linearly ordered by \(\subseteq\) andhas no maximal element then \(2|X| = |X|\). Howard/Yorke [1989] andNote 94. |
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| 3 H | \(E(Ia,V)\): For every set \(X\), if \(2|X| > |X|\) then \(X\)is amorphous. Howard\slash Yorke [1989], notes 85(H) and 94. |
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